Make three vertical cuts from top to bottom of each puff pastry sheet. The cuts should be about 2.5-inches (6-cm) apart. Repeat horizontally to make 18 squares.
Brush a square pastry lightly with water then place another square on top of it. Cut off any uneven edges. Repeat with the remaining squares.
Place the square pairs about 1-inch apart on a baking sheet, and brush the top of each with the egg wash.
Bake for 20 minutes or until pastry is a pale golden color. Remove from oven and let cool.
In a large pan, combine olive oil, peppers, salt and pepper. Cook for 8 minutes or until the peppers have softened.
Cut a rectangle in the middle of each square puff pastry and push it down. Fill the inside of the puff with a thin layer of bacon, cheddar cheese, and peppers. Use a small spoon or utensil to push the layers down. Carefully crack an egg on top.
Bake for an additional 20 minutes or until egg white is no longer translucent or runny.